Adam: The type of locks you'll need is dependent upon where you live. Here, in upstate New York, winters (generally October through April) are very cold, snowy, and they use salt to de-ice the highways. Department/hardware store locks will give you one winter season if kept clean from water, salt, and ice.
I would suggest purchasing high quality, stainless steel, and weather-proof padlocks, all keyed alike for each door AND the dual safety chain that normally is attached with a useless S-hook. Use one of those padlocks instead, replacing that S hook on safety chain.
Use another style high quality, stainless steel, and weather-proof padlock for the hitch lock that makes it very difficult to fit a hack saw or bolt cutter on the shackle arm.
When the locks are purchased brand new, shoot up the insides with a waterproofing lubricant similar to what pros use on heavy duty garage door springs. DO NOT use WD40 if regional temperatures drop below 32 degrees. WD40, though initially a cleaning agent, immediately becomes a crud magnet, and the crud, when frozen means the key slots freeze up too.
When the trailer is parked and disconnected from the towing vehicle, buy an extra towing ball and when you lock the hitch lock, put the spare ball into the hitch receiver. The eliminates someone towing it off using a smaller ball.
If additional security comfort is desired, get the "wheel club" or "wheel shoe". Those wire wheel locks are child's play to remove.
One other security tidbit. Any advertising on the exterior that relates to anything DJ (in some areas of our great land) translates directly to "Good stuff inside. If you have a drug habit, and are in need of quick cash, steal me!"
